On the existence of collective interactions reinforcing the metal-ligand bond in organometallic compounds

Recently, Sowlati-Hashjin et al. 1 concluded that the nature of the Li – C chemical bond in LiCF 3 differs signi ﬁ cantly from that in LiCPh 3 (Ph = phenyl). Whereas the Li – C bond of LiCF 3 is classi ﬁ ed as a conventional two-center two-electron bond (exchange-correlation interaction collectivity index, ICI XC = 0.910, ICI XC > 0.9 and close to 1), that of LiCPh 3 is categorized as a collective bond (ICI XC = 0.393). The authors claim that collective bonds take place in systems composed of MAR 3 (M = metal; A = C, B or Al; R = substituent) when M forms a stronger bond with the substituents R than with the central atom A. They claim the M – A interaction is either destabilizing or weakly stabilizing, whilst the 1,3-M (cid:129)(cid:129)(cid:129) R interactions are strongly stabilizing, but their method does not provide a causal mechanism that would demonstrate the correctness of this interpretation of the ICI XC index. Here, we prove the opposite, namely, that the Li – CPh 3 bond is not reinforced or provided by collective interactions, but that it is weakened by 1,3-M (cid:129)(cid:129)(cid:129) R contacts, which reduce the bond overlap. On top of that, there is 1,3-M (cid:129)(cid:129)(cid:129)

Recently, Sowlati-Hashjin et al. 1 concluded that the nature of the Li-C chemical bond in LiCF 3 differs significantly from that in LiCPh 3 (Ph = phenyl). Whereas the Li-C bond of LiCF 3 is classified as a conventional two-center two-electron bond (exchange-correlation interaction collectivity index, ICI XC = 0.910, ICI XC > 0.9 and close to 1), that of LiCPh 3 is categorized as a collective bond (ICI XC = 0.393). The authors claim that collective bonds take place in systems composed of MAR 3 (M = metal; A = C, B or Al; R = substituent) when M forms a stronger bond with the substituents R than with the central atom A. They claim the M-A interaction is either destabilizing or weakly stabilizing, whilst the 1,3-M•••R interactions are strongly stabilizing, but their method does not provide a causal mechanism that would demonstrate the correctness of this interpretation of the ICI XC index. Here, we prove the opposite, namely, that the Li-CPh 3 bond is not reinforced or provided by collective interactions, but that it is weakened by 1,3-M•••R contacts, which reduce the bond overlap. On top of that, there is 1,3-M•••R closed-shell overlap that further reduces the stability through Pauli repulsion. Taken together, our results suggest that there is no need to define the collective interaction as a new type of chemical bond.
We analyze the Li-C bond in LiCR 3 (R = F, Ph) using quantitative Kohn-Sham molecular orbital (MO) theory in conjunction with the activation strain model (ASM) and a matching energy decomposition analysis (EDA) 2-4 at M06-2X/TZ2P 5 with the Amsterdam Density Functional (ADF) 6,7 program. We stress that our physical model that provides causal relationships and thus explanations is the MO model and not, as often incorrectly stated, EDA. The latter is a tool that quantifies features in the MO bonding mechanism. Additionally, we analyze the Li-C bond in the doublet ground state of LiCR 2 • and the triplet ground state of LiCR •• using the optimized geometry of the parent molecule ( Supplementary Fig. 1). Table 1 gathers the results of the ASM and EDA for the homolytic Li-CR n (R = F, Ph; n = 1-3) bond cleavage into Li· + ·CR n radicals 8,9 . The ΔE oi component is the most important stabilizing contribution in the homolytic dissociation of LiCF 3 , which is driven by the bonding overlap S and energy difference Δε between the SOMOs of Li· and ·CF 3 ( Supplementary Figs. 2 and 3). In addition, there is only a weak donor-acceptor interaction between the lone pair on the F atoms and the SOMO and LUMO of Li ( Supplementary Fig. 4). Thus, the Li-C bond behaves as a typical electron-pair bond between Li· and ·CF 3 , and hence, is a polar covalent interaction. Interestingly, the Li-C bond becomes stronger going from LiCF 3 to LiCF 2 • to LiCF •• , due to stabilization of the SOMO(·CF n ) because reducing the number of F substituents reduces antibonding overlap between the C 2p z and F 2p z atomic orbitals (Supplementary Figs. 2 and 3). At difference, ΔV elstat slightly decreases from LiCF 3 to LiCF •• , by only around 3 kcal mol -1 .
Moving to LiCPh 3 , we find that the Li-C bond dissociation energy is around 20 kcal mol -1 less stabilizing than for LiCF 3 (Table 1). This weakening originates from the more destabilizing Pauli repulsion, due to the larger steric size and slightly shorter Li-R distance of R = Ph compared to R = F. For the same reason, ΔV elstat becomes more stabilizing, leading to a Li-C in the LiCPh 3 system with a nearly balanced ratio between the electrostatic (ΔV elstat ) and covalent (ΔE oi ) contributions. The ΔE oi , on the other hand, is similar to that of LiCF 3 ; whereas LiCPh 3 shows a weaker electron-pair bonding than LiCF 3 , this is compensated by stronger donor-acceptor interactions (Supplementary Table 4 and Supplementary Figs. [5][6][7][8]. These data indicate that the nature of the Li-C bond is similar in LiCF 3 and LiCPh 3 and do not favor a classification of the Li-C bond as conventional in LiCF 3 versus collective in LiCPh 3 . Additionally, if we had a collective bond in LiCPh 3 , we should expect a reduction of the Li-C bond strength when going from LiCPh 3  Notably, the in-phase overlap (Fig. 1, in purple) between the SOMO of Li (2s) and the SOMO of CR 3 (2p z fragments) to form the Li-C bond is much larger for LiCF 3 than for LiCPh 3 . At difference, the out-ofphase overlap (in orange) with the R groups is larger for the latter. Thus, the weaker Li-C interaction in LiCPh 3 , despite the expected collective interactions that should make it stronger, is in part also due to the cancelation of bond overlap that reduces the bond strength instead of providing bonding. The addition of more substituents in contact with Li drives to the cancellation of bond overlap as evidenced by the decrease in overlap densities from LiCPh 3 to LiCPh 2 • to LiCPh •• ( Supplementary Fig. 12). In other words, the collectivity of contacts reduces the bond strength instead of generating extra stability. This cancellation effect can also be observed in the comparison between LiCF 3 discussed above to i-LiCF 3 , which has also been considered to possess collective interactions ( Supplementary Fig. 11). Thus, the collectivity of contacts reduces the bond strength and hence affords no additional stability. This latter statement is also supported by the computed EDA-NOCV deformation densities ( Supplementary Fig. 18), which indicate similar orbital interactions in LiCF 3 , i-LiCF 3 , and LiCPh 3 .
In conclusion, our quantitative MO and EDA study does not reveal any sign of collective interaction in LiCPh 3 that makes the Li-C interaction stronger, but, in fact, the opposite. The small ICI XC , calculated as V XC (Li-C)/V XC (Li-{T}) where T stands for the set of all atoms of the system except the C directly attached to Li, of LiCPh 3 of Sowlati-Hashjin et al. 1 is likely the result of dividing a relatively low V XC (Li-C), because of the high polarity of this bond, by a large number of small long-range V XC (Li-C Ph ) and V XC (Li-H Ph ) contributions. It does not reflect any special chemical bond in this species, let alone a strong through-space interaction between the Li and the phenyl groups that could indicate collective bonding. Our results show that the nature of the Li-C bond of LiCPh 3 does not differ significantly from that of LiCF 3 , but only that the former has weaker electron-pair bonding, which is compensated by stronger donor-acceptor interactions. Finally, we have analyzed the inverted LiCF 3 cluster. Not unexpectedly, in this case, the ICI XC is small, but this is simply the result of a low V XC (Li-C) because of the large distance between Li and C and a large V XC (Li-F) due to the short Li-F distance (Supplementary Fig. 1 and Supplementary Table 2). Lastly, we anyway do not see a need to rebrand collective bonding as a new flavor of the chemical bond. The concept has been invoked already in other cases and is therefore not novel 10,11 .

Methods
All calculations were performed with the Amsterdam Density Functional (ADF) module of the AMS2021 software package at the M06-2X/ TZ2P level of theory [5][6][7]12 . The geometry optimizations were carried out without symmetry constraints (Supplementary Table 5). Numerical Hessians were computed to characterize the optimized structures as minima (zero imaginary frequencies). Geometries for LiCX 2 • and LiCX•• (X = F or Ph) were not allowed to relax but performed at the same geometry of LiCX 3 with the removal of either one (doublet state) or two X substituents (triplet state), respectively. LiCX 2 • and LiCX•• were computed at their doublet or triplet state, respectively. For this latter, the singlet state has been proven to be higher in energy in all cases.
The Li-C interaction was analyzed within the framework of quantitative Kohn-Sham molecular orbital theory in combination with a quantitative activation strain model (ASM) and energy decomposition analysis (EDA) in the gas phase. Both homolytic and heterolytic breaking schemes of Li-C bond have been studied. For instance, in LiCF 3 , we may have Li• (one unpaired alpha electron) and •CF 3 (one unpaired beta electron) fragments for the homolytic breaking. Or we may have Li + and -CF 3 fragments in case of the heterolytic breaking.

Activation strain and energy decomposition analysis
For the activation strain model (ASM), the bond energy ΔE between two fragments is made up of two components 3,13 : Here, the strain energy ΔE strain is the amount of energy required to deform the fragments from their equilibrium structure to the geometry that they acquire in the overall complex. The interaction energy Table 1 | Homolytic activation strain and energy decomposition analyses (in kcal mol -1 ) of LiCR n (R = F, Ph; n = 1-3), singlyoccupied molecular orbitals (SOMOs) energy difference (Δε in eV) and overlap integrals. a  ΔE int corresponds to the actual energy change when the geometrically deformed fragments are combined to form the overall complex. We further analyze the interaction ΔE int in the framework of the Kohn-Sham molecular orbital (MO) model, by dissecting it through our canonical energy decomposition analyses (EDA) into the electrostatic attraction, the Pauli repulsion and the (attractive) orbital interactions: The term ΔV elstat corresponds to the classical electrostatic interaction between the unperturbed charge distributions of the fragments in the geometry they possess in the complex. This term is usually attractive. The Pauli repulsion ΔE Pauli between these fragments comprises the destabilizing interactions, associated with the Pauli principle for fermions, between occupied orbitals and is responsible for the steric repulsion. The orbital interaction ΔE oi between these fragments in any MO model, and therefore also in Kohn-Sham theory, accounts for electron-pair bonding (the SOMO-SOMO interaction), charge transfer (empty/occupied orbital mixing between different fragments), and polarization (empty/occupied orbital mixing on one fragment due to the presence of another fragment). The orbital interaction energy ΔE oi can be further decomposed into the contributions from each irreducible representation Γ of the interacting system. The use of M06-2X gives a term that cannot be decomposed, which is a correction term, such that the total orbital interaction is the correct one.

Voronoi deformation density (VDD) charge
The electron density distribution is analyzed by using the Voronoi deformation density (VDD) method for atomic charges 14 . The VDD atomic charge Q VDD A is computed as the (numerical) integral of the deformation density Δρ(r) = ρ(r) -∑ B ρ B (r) in the volume of the Voronoi cell of atom A [Eq. (3)]. The Voronoi cell of atom A is defined as the compartment of space bound by the bond midplanes on and perpendicular to all bond axes between nucleus A and its neighboring nuclei (cf. the Wigner-Seitz cells in crystals) 14 .
In Eq. (3), ρ(r) is the electron density of the molecule and P B ρ B ðrÞ the superposition of atomic densities ρ B of a fictitious promolecule without chemical interactions that is associated with the situation in which all atoms are neutral. The interpretation of the VDD charge Q VDD A is rather straightforward and transparent. Instead of measuring the amount of charge associated with a particular atom A, Q VDD A directly monitors how much charge flows, due to chemical interactions, out of (Q VDD A > 0) or into (Q VDD A < 0) the Voronoi cell of atom A, that is, the region of space that is closer to nucleus A than to any other nucleus.

ETS-NOCV calculations
The natural orbitals for chemical valence method (ETS-NOCV) have been computed at the same M06-2X/TZ2P level of theory. It allows one to visualize the alteration in the electronic structure of the interacting species, which is associated with bond formation. In particular, we have depicted the deformation densities, whose shapes provide a visualization of the associated pairwise orbital interactions 15 .

Data availability
All data generated or analyzed during this study are included in this published article (and its supplementary information files).